A group is a structure G = (G,·,−1,e), where · is an infix binary operation, called
the group product, −1 is a postfix unary operation, called the
group inverse and e is a constant (nullary operation), called the
identity element, such that
· is associative: (xy)z = x(yz),
e is a left-identity for ·: ex = x, and
−1 gives a left-inverse: x−1x = e.
It follows that e is a right-identity and that −1gives a right
inverse: xe = x, xx−1 = e.
This definition shows that groups form a variety.
Let G and H be groups. A morphism from G to H is a function h : G→H that is a homomorphism: h(xy) = h(x)h(y) and h(x−1) = h(x)−1 and h(e) = e.
(SX,o,−1,idX), the collection of permutations of a sets X, with composition, inverse, and identity map.
The general linear group (GLn(V),·,−1,In), the collection of invertible n×n matrices over a vector space V, with matrix multiplication, inverse, and identity matrix.
|Equational theory||decidable in polynomial time|
|Congruence distributive||no (Z2×Z2)|
|Congruence n-permutable||yes, n=2, p(x,y,z) = xy−1z is a Mal'cev term|
|Congruence extension property||no, consider a non-simple subgroup of a simple group|
|Definable principal congruences|
|Equationally definable principal congruences||no|
|Strong amalgamation property||yes|
|Epimorphisms are surjective||yes|
Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html